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If the function f(x) is continous on the...

If the function f(x) is continous on the interval `[-2, 2],` find the values of a and b, where
`{:(f(x)(sin ax)/(x)-2",","for"-2lelt0),(=2x+1",", "for" 0lex le1),(=2b sqrt(x^(2)+3)-1",","for" 1 lt x le 2.):}`

Text Solution

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f is continous in `[-2,2]`
`therefore f` is continous at `x=0 and x=1.`
Continuity at `x=0`
`f(x)(sinax)/(x)-2"for"-2lex lt0`
`therefore underset(x to 0^(-))limf(x)=underset(x to0)lim((sinax)/(x)-2)`
`=underset(xto0)lim((sin ax)/(ax)xxa-2)`
`=aunderset(xto0)lim(sinax)/(ax)-2`
`=a(1)-2" "...[x to 0, ax to 0 and underset(theta to 0)lim (sintheta)/(theta)=1]=a-2`
`f(x)=2x+1,"for" 0 le x le1`
`therefoer f(0)=2(0)+1=1`
f is continous at `x=0`
`therefore underset(x to 0^(-))lim f(x)=f(0)`
`therefore a-2=1therefore a=3`
Continuity at `x=1`
From `(1),f(1)=1(1)+1=3`
`f(x)=2b sqrt(x^(2)+3)-1,` for `1 lt x le 2`
`thereforeunderset(x to 1^(+))lim f(x)=underset(x to 1)lim"("2bsqrt(x^(2)+3)-1")"`
`=2b underset(x to 1) limsqrt(x^(2)+3)-1`
`=2bsqrt(1+3)-1 =4b-1`
f is continous at `x=1`
`thereforeunderset(x to 1^(+))lim f(x)=f(1)`
`therefore4b-1=3`
`therefore4b=4`
`therefore bo=1`
`therefore` Hence, ` a=3, b=1.`
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